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Supplemental Material for

“Fully Consistent Density Functional Theory Determination of the Insulator-Metal Transition

Boundary in Warm Dense Hydrogen”

Joshua Hinz1, Valentin V. Karasiev1*, S. X. Hu1, Mohamed Zaghoo1, Daniel Mejía-Rodríguez2,

S.B. Trickey2, and L. Calderín3

*vkarasev@lle.rochester.edu

1 Laboratory for Laser Energetics, University of Rochester, Rochester, NY 14623

2 Quantum Theory Project, Department of Physics, University of Florida, Gainesville, FL 32611

3 Department of Materials Science and Engineering, University of Arizona, Tucson AZ 85721

06 Feb. 2020

Ab initio simulations:

Born-Oppenheimer molecular dynamic (BOMD) simulations were performed via the Vienna ab

initio simulation package (VASP) [1-3] using the SCAN-L [4,5] exchange-correlation (XC)

functional. The rVV10 [6,7] correlation correction was included in all simulations, unless

specifically stated otherwise. The motive is to incorporate the long range van der Waals

interaction that are not represented well by SCAN (or SCAN-L), despite its reasonable treatment

in the vicinity of equilibrium bond lengths. Those long-range contributions may be important

role in H2 dissociation. All simulations used the NVT ensemble on a system of 256 atoms in a

cubic super cell, with lattice parameters ranging from 7.05 to 10.27 Å. Electronic structures

were calculated at the Γ point with 256 bands. Convergence tests, see Figure S1, indicate that use

of Γ-point-only sampling introduces a maximum pressure uncertainty of 3%. Each MD

trajectory consists of 5000 to 6000 steps with a time step of 0.1 fs, giving a total simulation time

of 0.5 to 0.6 ps. The bare ion Coulomb potential was treated via the projector augmented wave

(PAW) framework [8] with the use of PBE PAWs (metaGGA PAWs are not available in VASP

at present).

Ring polymer path integral MD (PIMD) calculations for inclusion of nuclear quantum

effects (NQEs) were done with the i-PI [9,10] interface to Quantum Espresso (QE) [11,12].

Those simulations used the SCAN [13] XC functional with the rVV10 van der Waals correction.

These calculations used a local pseudo-potential [14] while the rest of the technical parameter

mailto:*vkarasev@lle.rochester.edu

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values were as consistent possible with those used in the BOMD simulations. All MD simulation

parameters are tabulated in Table S1 for reference.

With the exception of the 2500 and 3000 K IMT points, all other transition points were

obtained from simulations along various isochores with densities ranging from 0.8 to 1.15 g/cm3.

This procedure was chosen so as to follow thermodynamic paths consistent with those in the

static compression experiments. As a technical aside, as we moved along the isochore from low

to high temperature, we took the initial snapshot from the previous temperature point in an

attempt to converge the simulation faster by avoiding over-dissociated initial configurations.

Furthermore, the isotherms in the low pressure regime, which have a density range of 0.5 to 0.75

g/cm3, were chosen because of the unknown steepness of the IMT boundary slope at the outset of

this investigation.

Optical calculations:

The dynamic conductivity was calculated in the Kubo-Greenwood (KG) formalism [15,16] and

the dc conductivity extracted in the static field limit. All KG calculations in the main text

(BOMD and PIMD boundaries) were performed with VASP with the use of SCAN-L + rVV10

on a set of 20 evenly spaced snapshots from each MD trajectory. Note that the first 1000 steps of

each trajectory were skipped prior to taking the snapshots so as to allow for equilibration of the

simulation. The KG calculations used an automatically generated 2×2×2 Monkhorst-Pack k-

mesh with a plane-wave energy cutoff of 1000 eV and 256 bands. Results of KG convergence

tests for various thermodynamic conditions are shown in Figure S2. A potential inconsistency

may be introduced in the calculated PIMD IMT boundary as SCAN + rVV10 is used to produce

the ionic configurations and SCAN-L + rVV10 then is used to generate the Kohn-Sham orbitals

and eigenvalues for the calculated conductivity at each ionic configuration taken in the set of

snapshots. See discussion below regarding the small magnitude consequences of the

inconsistency.

The temperatures for which the average dc conductivity is directly above or below the

2000 S/cm criterion were identified. Linear interpolation between the two points then pinpointed

the transition temperature at which a dc conductivity of 2000 S/cm occurred. To estimate the

uncertainty in the calculated IMT temperature, the standard error of the dc conductivity was

added (or subtracted) from the dc conductivity for the two points in the fit and a reassessment of

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the transition temperature made. This procedure yields an estimated maximum uncertainty

associated with how the IMT was calculated, of ± 30 K. However, this is not a direct measure of

uncertainty from the theoretical approximations (primarily the XC approximation).

The corresponding reflectivity was calculated from the dynamical conductivity according

to reference [17]. Figure S3 shows the dc conductivity and reflectivity at 1.96 eV (calculated

relative to vacuum) for various isochores (some of which are not shown in the main text).

SCAN vs. SCAN-L:

As noted above and in the main text, the use of SCAN in the PIMD simulations and SCAN-L in

the KG calculations introduces inconsistency in the calculated PIMD IMT boundaries. SCAN-L

has been shown to reproduce, or nearly reproduce, various structural and energetic quantities

from SCAN for solids [5] and molecules [4]. The inconsistency arises in part because of the

inequivalence of the Kohn-Sham (KS) and generalized Kohn-Sham (gKS) schemes used to

determine the ground state electronic structure [4,5]. For reasons of complexity and

computational cost, SCAN calculations use gKS. SCAN-L calculations use KS. We did several

calculations to quantify the effects of that difference.

First we calculated hydrogen dc conductivities (with QE KGEC [18]) from SCAN +

rVV10 gKS orbitals and eigenvalues on the same set of ionic snapshots from the PIMD

trajectories that we used with SCAN-L + rVV10 KS orbitals and eigenvalues in the KG

calculations with VASP. The resulting transition temperature had an average increase of 11 K.

None of the transition points had a shift greater than 20 K. This shift is smaller than the estimated

methodological uncertainty; recall discussion above. As such the difference between SCAN and

SCAN-L (and the corresponding pseudo-potentials) inputs to the KG calculations of the PIMD

boundary is inconsequential.

The second issue regarding the consistency of SCAN versus SCAN-L is the matter of the

ionic trajectories and snapshots. That is, the comparison of PIMD and BOMD IMT boundaries

includes both the direct electronic structure distinctions between SCAN and SCAN-L but also

the possible direct NQEs. For a transition boundary that is clearly dependent on the ionic

configuration set, even minor changes in its generation might have a significant impact on the

IMT temperature. We quantify that effect next.

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Figure S4 shows a direct comparison of the ionic pair correlation function (PCF) from

SCAN-L + rVV10 and SCAN + rVV10 hydrogen BOMD simulations at 1.0 g/cm3 and 900 K. It

is clear that SCAN produces a system with a smaller molecular character (height of the first

peak) and molecules with a longer equilibrium bond length (position of first peak). Further

calculations of the dc conductivity along the 0.8, 0.9 and 1.0 g/cm3 isochores for BOMD

simulations are shown in Figure S5. Going from SCAN-L to SCAN to drive the BOMD

simulation shifts the IMT boundary higher in temperature by 5 to 7% for all three isochores.

Thus the shift in the IMT boundary associated with the inclusion of NQEs is underestimated by

roughly 100 K (effective cancellation of the two changes) when comparing the SCAN-L +

rVV10 BOMD results to the PIMD SCAN + rVV10 results.

PBE comparison 3000 K:

As can be seen in Figure 1 of the main text, the predicted IMT boundary from SCAN-L + rVV10

(BOMD) appears to approach that from PBE around 3000 K. To investigate this behavior

further, the dc conductivities and the height of the first peak of the PCF from PBE and SCAN-

L+rVV10 (both classical nuclei) along the 2500 K and 3000 K isotherms are compared in Figure

S6. At 3000 K, one sees clearly that the difference between PBE and SCAN-L + rVV10 results

for both properties becomes significantly smaller than at lower temperatures across the whole

density (or pressure) range of the isotherm. Furthermore, there is a striking similarity to the

results from the two functionals for the overall structure of the curves of the two properties at

3000 K that does not appear in the comparison along the 2500 K isotherms. This finding helps

support the notion that the predicted IMT boundaries of SCAN-L + rVV10 and PBE do in fact

become close above 3000 K and that the two XC functionals predict a fluid hydrogen system

with a similar level of accuracy at such thermodynamic conditions.

References:

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semiconductors using a plane-wave basis set. Comput. Mater. Sci. 6, 1550 (1996).

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4. D. Mejía-Rodríguez, S. B. Trickey, Deorbitalization strategies for meta-generalized-

gradient-approximation exchange-correlation functionals. Phys. Rev. A 96, 052512

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the better. J. Chem. Phys. 133, 244103 (2010).

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semilocal density functional. Phys. Rev. Lett. 115, 036402 (2015).

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noninteracting free energy functionals for orbital free density functional calculations,

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16. D. A. Greenwood, The Boltzmann equation in the theory of electrical conduction in

metals. Proc. Phys. Soc. Lond. 71, 585596 (1958).

17. S. X. Hu et al., Impact of first-principles properties of deuterium-tritium on inertial

confinement fusion target designs. Phys. Plasmas 22, 056304 (2015).

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Fig. S1. Convergence tests performed by single point calculations for the setup of the MD

simulation parameters including the plane –wave energy cutoff, k-mesh and number of bands.

All red curves correspond to the free energy of the system (not including contribution from the

ionic entropy) and blue curves correspond to the total pressure.

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Fig S2. Single snapshot calculations of the dc conductivity as a function of k points (top left and

right and bottom left) and plane-wave energy cutoff (bottom right) used to determine the

converged set of simulation parameters for the dc conductivities calculations over the set of

snapshots for each thermodynamic condition.

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Fig S3. The dc conductivity (top) and reflectivity (bottom) of the PIMD isochores with the use of

SCAN-L + rVV10 in the KG calculation. Note the horizontal dotted black line in the dc

conductivity panel indicates the 2000 S/cm criterion used to determine the IMT.

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Fig. S4. Pair correlation function (PCF) for 4 separate simulations of hydrogen at 1.0 g/cm3 and

900 K. Blue curve corresponds to the BOMD simulation with VASP using SCAN-L + rVV10.

Red curve is the PCF from PIMD simulation with one bead (i.e. BOMD) using SCAN + rVV10.

Green and black are PCFs from PIMD simulations with eight and four beads respectively.

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Fig. S5. Comparison of dc conductivities along the 0.8 (top), 0.9 (middle) and 1.0 g/cm3

(bottom) isotherms. The dotted black line indicates a dc conductivity of 2000 S/cm. Each red

curve corresponds to the BOMD simulation with the use of SCAN-L + rVV10 and the green

curve corresponds to BOMD simulations with SCAN + rVV10. Note that the KG calculations

for both sets of data were performed with SCAN-L + rVV10.

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Fig S6. Left, comparison of the dc conductivity along the 2500K and 3000K isotherms for

SCAN-L + rVV10 and PBE (BOMD). Right, first peak of the pair correlation function for the

corresponding dc conductivity curves in the left panel.

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Parameter BOMD PIMD

# atoms 256 256

Plane-Wave cutoff

energy

750 eV 2700 eV

Convergence criterion

on the total electronic

free energy

1E-5 eV 2.7E-5 eV

# bands 168 160

K pt’s gamma gamma

Time step 0.1 fs 0.2 to 0.5 (adjust for

temperature and density)

# MD steps 6000 6000

Electron partial

occupancies

Fermi smearing Fermi smearing

Bare proton treatment Projector Augmented

Wavepotentials [8]

Local [14]

Ensemble NVT NVT

Thermostat Nose-Hoover (acts every 40 MD

steps)

Langevin (acts every 200

fs)

# beads N.A. 8

Table S1. Quick reference for the parameters used in the BOMD and PIMD simulations. Note,

in the case of Fermi smearing the smearing temperature is set to the ionic temperature to be

maintained by the thermostat.